Why is it that when I say "I will give this book to my daughter", I am using the verb "give" and the preposition "to", but "to" is not used in the following: "What kind of names do people in your culture give pets?"

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It sounds like you need basic instruction in indirect objects and basic SVIO word-order in English. We have lots of questions about those here, too. You may wish to check out our sister site for English Language Learners. It’s “for people who are learning or teaching English as a foreign language”, whereas here we’re more intended “for linguists, etymologists, and (serious) English language enthusiasts”
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tchristAug 16 '13 at 1:33

1 Answer
1

English can be a flexible language at times, believe it or not. Any of the following sentences is correct grammatically:

1) What kind of names do people in your culture give pets?

2) What kind of names do people in your culture give to pets?

3) I will give my daughter this book.

4) I will give to my daughter this book.

5) I will give this book to my daughter.

And there are probably many other combinations and permutations of word order that I could give you (or give to you), but I'll stop at five. As for the reason why your two sentences have to be worded differently, I'll leave that to the grammarians.